In this paper, we propose two new explicit Almost Runge-Kutta (ARK) methods, ARK3 (a three stage third order method,
i.e.,
s =
p = 3) and ARK34 (a four-stage third-order method,
i.e.,
s = 4,
p = 3), for the numerical solution of initial value problems (IVPs). The methods are derived through the application of order and stability conditions normally associated with Runge-Kutta methods; the derived methods are further tested for consistency and stability, a necessary requirement for convergence of any numerical scheme; they are shown to satisfy the criteria for both consistency and stability; hence their convergence is guaranteed. Numerical experiments carried out further justified the efficiency of the methods.
Almost Runge-Kutta Stability Consistency Convergence Order Conditions Rooted Trees1. Introduction
According to [1] the s-stage Runge-Kutta method for solving the initial value problem
is defined by
where
and
Alternative forms of the above equations are:
where
The two forms of Equations (2) and (5) are equivalent by making the interpretation
where is the inner stages that tend to estimate the solutions at some points; s is the number of stages and is the points where the function f is computed for a step. ARK methods are a special class of RK methods that arose out of the quest to develop efficient and accurate methods that have advantages over the traditional methods by retaining the simple stability function of RK methods, allowing minimal information to be passed between steps and adjusting stepsize easily. Since the introduction of ARK methods in by [2] , other researchers who have made their input toward the development of this method include [3] - [7] .
2. Materials and Methods2.1. Method ARK3 (s = p = 3)
The general third order three stages Almost Runge-Kutta scheme is of the form:
We represent the abscissa vector.
The order conditions for order three ARK schemes are derived through the standard rooted tree approach used for Runge-Kutta methods [8] .
The conditions of Runge-Kutta stability for 3rd order, 3 stages are:
where.
Acquiring order 2 estimation with respect to 2nd scaled derivative for the 3rd outgoing solution, we need:
From Equation (12) we have,
Solving Equation (9) we obtain
And from Equation (11), we obtain
Evaluating both sides of Equation (10) we obtain
This implies that
Thus Equation (13) becomes
Two free parameters, and are required for an order three scheme. Thus; and after
calculating the members of the U matrix we obtain the a scheme for method.
2.2. Method ARK34 (s = 4, p = 3)
The third order four stages scheme has the general form:
Its stability function is expressed as
The order conditions are derived using the standard rooted tree approach used for Runge-Kutta methods [8] .
The values are obtained by expanding
Also,
There is also the additional condition
and L will be assumed to be the free parameters, where is the error coefficient comparable to the bushy tree. From Equations (25)-(27) together with Equation (34) we have
Thus
From Equation (30) we obtain
Evaluating the stability matrix of a four stage third order method, we arrive at
where Tr is the trace of a matrix and
Hence,
And it follows that:
Since we obtain
We introduce, and. Thus from Equation (46) we arrived at
And from Equations (32) and (33) we obtain respectively
Further simplification produces the following results
Setting and substituted this into Equation (29), we obtain
Thus
And the proposed ARK34 with is
3. Convergence Analysis
For the method ARK3 represented by Equation (24), the matrix
must have bounded powers for the method to be stable.
The characteristic polynomial of V is given as
Thus
Applying Cayley-Hamilton theorem to matrix V
This implies that
Similarly,
for every n greater than 2. It implies is bounded, which shows that the method is stable. It is known that methods of order at least one are always consistent; hence the method is consistent since the order of the method is. Therefore, Hence the proposed scheme ARK3 is convergent due to the fact that it is both stable and consistent.
Similarly, for the ARK34 method of Equation (61), the matrix
And the eigenvalues are evaluated to be.
Thus,
And similarly, it implies that, for every n greater than 2. It indicates that is bounded which shows that the method is stable. Also, the method is consistent since it is of order 3, i.e.,. Hence the proposed scheme (ARK34) is convergent due to the fact that it is both stable and consistent.
4. Numerical Examples
Considering the problem below:
Source: Rattenbury [3] .
Problem (72) is solved using the proposed ARK34 method. The results are obtained and compared with similar ARK34 methods of [3] and [5] respectively and presented in Figure 1.
Comparison of ARK34 with other methods (h = 0.1)
From Figure 1 it is evident that our Proposed ARK34 method performed better than the methods of [3] and [5] since it exhibits lesser error than the errors of the existing methods.
5. Conclusion
Two ARK methods are proposed, ARK3 and ARK34. The methods have been proven to be consistent and stable, thereby guaranteeing their convergence. This is further illustrated by comparing the performance of one of the methods with other methods of similar order. The proposed method ARK34 is shown to perform better than the existing methods.
Acknowledgements
The authors would like to thank the reviewer(s) for their constructive criticisms.
Cite this paper
AbdulrahmanNdanusa,Khadeejah JamesAudu, (2016) Design and Analysis of Some Third Order Explicit Almost Runge-Kutta Methods. Applied Mathematics,07,13-21. doi: 10.4236/am.2016.71002
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